Lemma 51.17.2. Let $A$ be a ring. If $f_1, \ldots , f_{r - 1}, f_ rg_ r$ are independent, then $f_1, \ldots , f_ r$ are independent.

See [Lech-inequalities] and [Lemma 1 page 299, MatCA].

**Proof.**
Say $\sum a_ if_ i = 0$. Then $\sum a_ ig_ rf_ i = 0$. Hence $a_ r \in (f_1, \ldots , f_{r - 1}, f_ rg_ r)$. Write $a_ r = \sum _{i < r} b_ i f_ i + b f_ rg_ r$. Then $0 = \sum _{i < r} (a_ i + b_ if_ r)f_ i + bf_ r^2g_ r$. Thus $a_ i + b_ i f_ r \in (f_1, \ldots , f_{r - 1}, f_ rg_ r)$ which implies $a_ i \in (f_1, \ldots , f_ r)$ as desired.
$\square$

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